1SLARRE(1) LAPACK auxiliary routine (version 3.1) SLARRE(1)
2
6 SLARRE - the desired eigenvalues of a given real symmetric tridiagonal
7 matrix T, SLARRE sets any "small" off-diagonal elements to zero, and
8 for each unreduced block T_i, it finds (a) a suitable shift at one end
9 of the block's spectrum,
10
12 SUBROUTINE SLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2, RTOL1, RTOL2,
13 SPLTOL, NSPLIT, ISPLIT, M, W, WERR, WGAP, IBLOCK,
14 INDEXW, GERS, PIVMIN, WORK, IWORK, INFO )
15 IMPLICIT NONE
17 CHARACTER RANGE
19 INTEGER IL, INFO, IU, M, N, NSPLIT
21 REAL PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU
23 INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ), INDEXW( * )
25 REAL D( * ), E( * ), E2( * ), GERS( * ), W( * ),WERR( *
27 ), WGAP( * ), WORK( * )
28
30 To find the desired eigenvalues of a given real symmetric tridiagonal
31 matrix T, SLARRE sets any "small" off-diagonal elements to zero, and
32 for each unreduced block T_i, it finds (a) a suitable shift at one end
33 of the block's spectrum, (b) the base representation, T_i - sigma_i I =
34 L_i D_i L_i^T, and (c) eigenvalues of each L_i D_i L_i^T.
35 The representations and eigenvalues found are then used by SSTEMR to
36 compute the eigenvectors of T.
37 The accuracy varies depending on whether bisection is used to find a
38 few eigenvalues or the dqds algorithm (subroutine SLASQ2) to conpute
39 all and then discard any unwanted one.
40 As an added benefit, SLARRE also outputs the n
41 Gerschgorin intervals for the matrices L_i D_i L_i^T.
42
45 RANGE (input) CHARACTER
46 = 'A': ("All") all eigenvalues will be found.
47 = 'V': ("Value") all eigenvalues in the half-open interval (VL,
48 VU] will be found. = 'I': ("Index") the IL-th through IU-th
49 eigenvalues (of the entire matrix) will be found.
50 N (input) INTEGER
52 The order of the matrix. N > 0.
53 VL (input/output) REAL
55 VU (input/output) REAL If RANGE='V', the lower and upper
56 bounds for the eigenvalues. Eigenvalues less than or equal to
57 VL, or greater than VU, will not be returned. VL < VU. If
58 RANGE='I' or ='A', SLARRE computes bounds on the desired part
59 of the spectrum.
60 IL (input) INTEGER
62 IU (input) INTEGER If RANGE='I', the indices (in ascending
63 order) of the smallest and largest eigenvalues to be returned.
64 1 <= IL <= IU <= N.
65 D (input/output) REAL array, dimension (N)
67 On entry, the N diagonal elements of the tridiagonal matrix T.
68 On exit, the N diagonal elements of the diagonal matrices D_i.
69 E (input/output) REAL array, dimension (N)
71 On entry, the first (N-1) entries contain the subdiagonal ele‐
72 ments of the tridiagonal matrix T; E(N) need not be set. On
73 exit, E contains the subdiagonal elements of the unit bidiago‐
74 nal matrices L_i. The entries E( ISPLIT( I ) ), 1 <= I <=
75 NSPLIT, contain the base points sigma_i on output.
76 E2 (input/output) REAL array, dimension (N)
78 On entry, the first (N-1) entries contain the SQUARES of the
79 subdiagonal elements of the tridiagonal matrix T; E2(N) need
80 not be set. On exit, the entries E2( ISPLIT( I ) ), 1 <= I <=
81 NSPLIT, have been set to zero
82 RTOL1 (input) REAL
84 RTOL2 (input) REAL Parameters for bisection. RIGHT-
85 LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
86 SPLTOL (input) REAL The threshold for splitting.
88 NSPLIT (output) INTEGER
90 The number of blocks T splits into. 1 <= NSPLIT <= N.
91 ISPLIT (output) INTEGER array, dimension (N)
93 The splitting points, at which T breaks up into blocks. The
94 first block consists of rows/columns 1 to ISPLIT(1), the second
95 of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the
96 NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through
97 ISPLIT(NSPLIT)=N.
98 M (output) INTEGER
100 The total number of eigenvalues (of all L_i D_i L_i^T) found.
101 W (output) REAL array, dimension (N)
103 The first M elements contain the eigenvalues. The eigenvalues
104 of each of the blocks, L_i D_i L_i^T, are sorted in ascending
105 order ( SLARRE may use the remaining N-M elements as
106 workspace).
107 WERR (output) REAL array, dimension (N)
109 The error bound on the corresponding eigenvalue in W.
110 WGAP (output) REAL array, dimension (N)
112 The separation from the right neighbor eigenvalue in W. The
113 gap is only with respect to the eigenvalues of the same block
114 as each block has its own representation tree. Exception: at
115 the right end of a block we store the left gap
116 IBLOCK (output) INTEGER array, dimension (N)
118 The indices of the blocks (submatrices) associated with the
119 corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i)
120 belongs to the first block from the top, =2 if W(i) belongs to
121 the second block, etc.
122 INDEXW (output) INTEGER array, dimension (N)
124 The indices of the eigenvalues within each block (submatrix);
125 for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the i-th
126 eigenvalue W(i) is the 10-th eigenvalue in block 2
127 GERS (output) REAL array, dimension (2*N)
129 The N Gerschgorin intervals (the i-th Gerschgorin interval is
130 (GERS(2*i-1), GERS(2*i)).
131 PIVMIN (output) DOUBLE PRECISION
133 The minimum pivot in the Sturm sequence for T.
134 WORK (workspace) REAL array, dimension (6*N)
136 Workspace.
137 IWORK (workspace) INTEGER array, dimension (5*N)
139 Workspace.
140 INFO (output) INTEGER
142 = 0: successful exit
143 > 0: A problem occured in SLARRE.
144 < 0: One of the called subroutines signaled an internal prob‐
145 lem. Needs inspection of the corresponding parameter IINFO for
146 further information.
147 =-1: Problem in SLARRD.
149 = 2: No base representation could be found in MAXTRY iterations.
150 Increasing MAXTRY and recompilation might be a remedy. =-3:
151 Problem in SLARRB when computing the refined root representation
152 for SLASQ2. =-4: Problem in SLARRB when preforming bisection on
153 the desired part of the spectrum. =-5: Problem in SLASQ2.
154 =-6: Problem in SLASQ2.
155 Further Details element growth and consequently define all their
157 eigenvalues to high relative accuracy. ===============
158 Based on contributions by Beresford Parlett, University of Cali‐
160 fornia, Berkeley, USA Jim Demmel, University of California,
161 Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA
162 Osni Marques, LBNL/NERSC, USA Christof Voemel, University of Cal‐
163 ifornia, Berkeley, USA
164 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 SLARRE(1)